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Degeneration of a graph describing conformal structure A. B. Bogatyrev Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 3, DOI: https://doi.org/10.4213/sm9703 
Bibliographic databases:
    § 1. IntroductionRiemann surfaces do not necessarily appear as algebraic curves. Other representations are much more convenient for some applications. For instance, one can glue a surface of some standard pieces of the complex plane, half-planes, triangles, rectangles, (half-) strips and so on. The scars remaining after this surgery make up a graph embedded in the surface. The idea to represent complex structures and even more delicate objects on surfaces (like abelian differentials, quadratic differentials [1], including Jenkins-Strebel ones, branched projective structures [2] and so on) by embedded weighted graphs is not new. Klein [3], [4] was perhaps the inventor of this tradition. The most prominent examples of this kind are dessins d’enfants introduced by Grothendieck [5], [6]. Similar constructions were used by Bertola in his work on Boutroux curves [7]. A triangulation of the Teichmüller space of a punctured surface on the basis of ideas due to Thurston was performed by several authors — see, for instance, the survey [8] by Harer — and was aimed at the investigation of the topology of the moduli space. The critical graphs of quadratic differentials, which appear in Mumford’s construction of the arc complex of a surface, were used subsequently by Kontsevich, under the brandname of ‘ribbon graphs’, to build the intersection theory on moduli spaces (see [9]). Eventually, this work led to the first proof of Witten’s conjecture; see [6] and [10]–[12]. The investigation of the period map which lies in the core of the Chebyshev Ansatz method used for the solution of problems in uniform rational approximation [13], [14] is also grounded on graphical techniques [15], [16]. Other closely related topics include flat surfaces [17], triangulated surfaces [18], the complex Pell-Abel equations [19]; also see [20] and [21] for other examples. Graphical techniques proved to be extremely useful in the study of the mesoscale geometry (as opposed to differential and global geometry) of various moduli spaces and related structures. Graphs describing conformal structures on a surface can change their combinatorial characteristics, like in flip transformations of a ribbon graph. It is intuitively clear that the complex structure of the underlying surface should have a continuous limit as we contract edges of the graph. However, a detailed mathematical analysis of what happens to moduli under this transformation has not been done yet — to the best of our knowledge. The response of the moduli of a curve to variations of weights of the graph strictly within the space of admissible weights was investigated long ago. This dependence is real analytic: see, for example, [14] and [15]. The same problem near the boundary of the space of admissible weights is much more difficult since we have to compare the moduli of surfaces glued in accordance with different rules: this reflects the change in the combinatorial structure of the graph. In this paper we consider a very concrete example: the moduli space $\mathcal{H}_2^1$ of genus $2$ real curves with unique real oval and a marked point on it which is disjoint from the ramification points. This space can be decomposed into nine full-dimensional cells labelled by special trees shown in Figure 1. We consider a transition across the wall separating two neighbouring cells and show that the natural moduli of curves given by the positions of the branch points behave continuously but not smoothly with respect to the natural coordinates in each cell: a root-type singularity arises at the wall. In particular, the main result provides a justification for the adjacency relations for cell decompositions in the combinatorial theory of moduli spaces, which is often done (especially by physicists) at the intuitive level, without a deep analysis, which can be rather sophisticated. The main method of investigation we use is a quasiconformal deformation [22] of abelian integrals — a simplified version of the technique proposed in [14], Ch. 5. The quasiconformal approach is a traditional and a universal tool for the investigation of various coordinate systems in moduli spaces, including a neighbourhood of a singular point of a change of coordinates. The choice of the moduli space $\mathcal{H}_2^1$ as an object of study is accidental to a certain extent: this space is neither elementary, nor too involved. What is also important, this space is used in the solution of some applied problems [23], [24], [14]. The main ingredients of our analysis can be extended to much more sophisticated cases, including, for instance, merging simple zeros of the distinguished differential. AcknowledgementsThe author thanks the participants of A. Gonchar’s Seminar on complex analysis (Steklov Mathematical Institute) and G. Shabat’s seminar Graphs on surfaces and curves over number fields (Moscow State University) for discussions related to this paper and for their constructive criticisms. My special gratitude goes to Prof. Hartmuth Monien for drawing my attention to the works [3] and [4] by F. Klein. The author is also grateful to anonymous referees, who contributed to making the presentation more transparent. § 2. Preliminary settings and the main resultIn this section we recall the definition of the moduli space of real genus $2$ curves with exactly one oriented oval and a marked point on it; we present the description of curves in terms of weighted graphs and formulate the main result of this paper. 2.1. The moduli spaceThere are two classes of smooth genus $2$ real curves with unique real oval: the hyperelliptic involution $J$ (acting on each curve) either preserves the orientation of the oval or reverses it. Let $\mathcal{H}_2^1$ be the moduli space of curves $M$ in the second class which we equip additionally with an orientation of the real oval and a marked point ‘$\infty$’ on it (we drop upper commas in what follows). This marked point should not be fixed by the involution $J$. This moduli space is used, for example, in the analysis of the problem of so-called optimal stability polynomials (see [23], [24] and [14]), including the case of damped polynomials. Any element of $\mathcal{H}_2^1$ admits a normalized affine model:
$$
\begin{equation}
M=M(\mathsf E):= \biggl\{(x,w)\in\mathbb{C}^2\colon w^2=(x^2-1) \prod_{s=1}^2(x-e_s)(x-\overline{e}_s) \biggr\}
\end{equation}
\tag{2.1}
$$
with branch points $e_1$ and $e_2$, $e_1\neq e_2$, lying in the open upper half-plane $\mathbb{H}$. Its branch locus $\mathsf{E}=\{\pm1, e_s,\overline{e}_s\}_{s=1,2}$ is mirror symmetric: $\mathsf E=\overline{\mathsf E}$. We define the hyperelliptic and anticonformal involutions on the curve by $J(x,w):=(x,-w)$ and $\overline{J}(x,w):=(\overline{x},\overline{w})$, respectively. The marked point $\infty$ on the real oval is the point corresponding to $(x,w)=(+\infty,+\infty)$ in the natural two-point compactification of the affine curve (2.1). The space $\mathcal{H}_2^1$ of such curves is parametrized by the positions of their branch points $e_1$ and $e_2$ in the open upper half plane $\mathbb{H}$, which are indistinguishable and cannot coincide. Therefore, we have a model $(\mathbb{H}^2\setminus\{\text{diagonal}\})/\text{permutation}$ for the moduli space, which shows that $\dim\mathcal{H}_2^1=4$ and $\pi_1(\mathcal{H}_2^1)=\mathrm{Br}_2=\mathbb{Z}$. 2.2. The distinguished differentialOn each curve $M$ in the moduli space there is a unique abelian differential of the third kind $d\eta_M$ with just two simple poles, the marked point $\infty$ and its involution $J\infty$, with residues $-1$ and $+1$ at these points, respectively, and with purely imaginary periods (see [14], § 2.1.3). For the algebraic model (2.1) of $M$ this differential takes the form where dots denote a linear polynomial in the variable $x$. One can check that the normalization conditions on the differential imply that it is real, that is, ${\overline{J}d\eta_M=\overline{d\eta_M}}$. In other words, the linear polynomial in (2.2) has real coefficients. An important consequence of this fact is as follows: the periods of this differential along even 1-cycles $C:=\overline{J}C$ vanish since they should be real and imaginary at the same time (see [14] and [13]). We can express the solutions of various problems in uniform polynomial approximation [14] in terms of just this differential, namely,
$$
\begin{equation*}
P_n(x)=\cos\biggl(ni\int_{(1,0)}^{(x,w)}\,d\eta_M\biggr)
\end{equation*}
\notag
$$
under the additional condition $\displaystyle\int_{H_1(M,\mathbb{Z})}\!\!\,d\eta_M\subset\frac{2\pi i}n\mathbb{Z}$, which means that the periods of the abelian integral are commensurable with the periods of the cosine and therefore guarantees that the left-hand side of the equality is a polynomial. Classical formulae for Chebyshev and Zolotarev polynomials are just particular cases of this representation. 2.3. The global width functionLet $M(\mathsf{E})\in\mathcal{H}_2^1$, and let $d\eta_M$ be the distinguished differential associated with $M$ as above. We immediately check that the normalization conditions on $d\eta_M$ imply that the width function
$$
\begin{equation}
W(x):=\biggl|\operatorname{Re}\int_{(1,0)}^{(x,w)}\,d\eta_M\biggr|, \qquad x\in\mathbb{C},
\end{equation}
\tag{2.3}
$$
has the following properties:
We only comment on the last property. Since $d\eta_M$ is odd with respect to the hyperelliptic involution of $M$, $W(e_s)$ is equal to half the absolute value of the real part of some period of the differential. By definition, all its periods are purely imaginary. The level lines of $W$ form the vertical foliation associated with the quadratic differential $(d\eta)^2$, whereas its steepest descent lines are horizontal trajectories of this differential. 2.4. The construction of the graph $\Gamma(M)$To any curve $M$ in our moduli space we assign a weighted planar graph $\Gamma=\Gamma(M)$ which is formed by a finite number of segments of the vertical and horizontal foliations [1] associated with the quadratic differential $(d\eta_M)^2$ pushed forward to the $x$-plane. The graph $\Gamma(M)$ is the union of the ‘vertical’ subgraph $\Gamma_ {\rule{.5mm}{2.mm}} $ and the ‘horizontal’ subgraph $\Gamma_ {\rule{2.mm}{.5mm}} $; see Figures 1 and 2 for examples of admissible graphs (obtained from some curve $M$). Definition 1. Vertical edges of the graph are arcs of the zero set of $W(x)$; they are segments of the vertical foliation $(d\eta_M)^2<0$ and are not oriented. The horizontal edges are the segments of the horizontal foliation $(d\eta_M)^2>0$ (or the steepest descent lines of $W(x)$) that connect saddle points of $W$ with its zero set (and pass occasionally through other saddle points). Horizontal edges are oriented in the increasing direction of $W(x)$. Each edge, of no matter which type, is equipped with the weight equal to its length in the metric $ds=|d\eta_M|$ of the quadratic differential. The vertices of the graph $\Gamma$ include all finite points in the divisor of the quadratic differential $({d\eta_M})^2$ considered on the plane, as well as the points in $\Gamma_ {\rule{.5mm}{2.mm}} \cap\Gamma_ {\rule{2.mm}{.5mm}} $, which are the projections of saddle points of $W$ onto its zero set along horizontal leaves. Remark 1. Instead of assigning lengths to horizontal edges, it is more convenient to keep the values of the width function $W(x)$ at all vertices of the graph: then the length of an oriented edge is the increment of the width function along it. From the local behaviour of trajectories one immediately sees that the multiplicity of a vertex $V$ of the graph in the divisor of the quadratic differential $(d\eta_M)^2$ is equal to a combinatorial quantity:
$$
\begin{equation*}
\operatorname{ord} (V):= d_ {\rule{.5mm}{2.mm}} (V)+2d_{\mathrm{in}}(V)-2,
\end{equation*}
\notag
$$
where $d_ {\rule{.5mm}{2.mm}} $ is the degree of the vertex with respect to the vertical edges and $d_{\rm in}$ is the number of incoming horizontal edges. Branch points of the curve correspond to vertices $V$ of the subgraph $\Gamma_ {\rule{.5mm}{2.mm}} \subset\Gamma$ with odd value of $\operatorname{ord}(V)$; see Figure 2 for an example. A weighted planar graph can be associated with an arbitrary (real) hyperelliptic curve with marked point on it (on its oval) as described above; see [19]. Admissible graphs can be characterized in an axiomatic way: there are five restrictions on the combinatorics and weights of a graph, which are explicitly listed in [15] and [14] and characterize such graphs exhaustively. These axioms can be actualized by means of a combinatorial algorithm which lists all admissible graphs. 2.5. The coordinate space of a graphIt turns out that in the case of the moduli space $\mathcal{H}_2^1$ there are only nine ‘stable’ topological types of graphs $\Gamma$ which do not change their combinatorial structure under arbitrary small perturbations of a curve $M(\mathsf{E})$; these are listed, for instance, in [16]. Two of such graphs $\Gamma_+$ and $\Gamma_-$ are shown in Figure 2. The admissible independent weights sweep a four-dimensional space $\mathcal{A}[\Gamma]$, called the coordinate space of the graph. These spaces have an explicit description:
$$
\begin{equation*}
\mathcal A[\Gamma_-]=\bigl\{(H_1,H_2,W_1,W_2)\in\mathbb{R}_+^4\colon 2(H_1+H_2)<\pi;\, W_1<W_2\bigr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal A[\Gamma_+]=\bigl\{(H_0,H_1,H_2,W)\in\mathbb{R}_+^4\colon H_0+2(H_1+H_2)<\pi\bigr\}, \qquad \mathbb{R_+}:=(0,\infty)
\end{equation*}
\notag
$$
(see [15] and [14]). Each coordinate space is the interior of the product of a simplex swept by the variables $H$ by a cone swept by the variables $W$. Along with the topologically stable types of graphs $\Gamma$, there are lots of intermediate unstable ones, like the graph $\Gamma_0$ shown in Figure 3. Its coordinate space has codimension $1$ in the moduli space $\mathcal{H}_2^1$:
$$
\begin{equation*}
\mathcal A[\Gamma_0]=\bigl\{(H_1,H_2,W)\in\mathbb{R}_+^3\colon 2(H_1+H_2)<\pi\bigr\},
\end{equation*}
\notag
$$
and, embedded in the moduli space, it serves as an interface between the coordinate spaces of $\Gamma_\pm$. Each of the graphs $\Gamma_-$, $\Gamma_0$ and $\Gamma_+$ with weights in the relevant coordinate space can be realized as the graph of a single curve $M$ in the moduli space $\mathcal{H}_2^1$ (see [15] and [14]). This Riemann surface can be glued from a finite number of strips in a way determined by the combinatorics and weights of the graph. Detailed instructions for gluing were given in [14]–[16]. Unfortunately, this approach to the reconstruction of a curve cannot be called effective, and it does not give us any quantitative characteristics of the embedding of the coordinate space in the moduli space. We use more flexible and constructive methods related to quasiconformal mappings in what follows. This author claimed in [16] that the embedding of a coordinate space of positive codimension like $\mathcal{A}[\Gamma_0]$ coincides with the continuations of the embeddings of suitable full-dimensional polyhedra $\mathcal{A}[\Gamma]$ to faces of these polyhedra. Here we verify this in a particular case. The embedding of a coordinate space in a moduli space (see [14] and [15]) will be given by a pair of complex-valued real analytic functions $e_s(H,W)$, $s=1,2$, defined in the interior of $\mathcal{A}[\Gamma]$ and describing the dependence of the branch divisor of the curve on the weights of the graph $\Gamma$. In this paper we study the behaviour of these functions near the boundary $\{W_1=W_2\}$ of the coordinate space $\mathcal{A}[\Gamma_-]$ and the boundary $\{H_0=0\}$ of the polyhedron $\mathcal{A}[\Gamma_+]$. 2.6. The main theoremConsider an arbitrary point $A^0:=(H_1,H_2,W)$ in the coordinate space $\mathcal{A}[\Gamma_0]$ of codimension $1$ and its small displacement $\delta A:=(\delta H_1, \delta H_2, \delta W)$ within this space. A ‘transversal’ displacement to the adjoining full-dimensional spaces $\mathcal{A}[\Gamma_\pm]$ will be described by a (small) positive variable $h$. In combination with the tangential shift $\delta A$, it defines two points,
$$
\begin{equation}
\begin{aligned} \, \notag \mathcal A[\Gamma_-]\ni A^- &=(H_1^-,H_2^-,W_1^-,W_2^-) \\ &:=(H_1+\delta H_1,\,H_2+\delta H_2,\,W+\delta W-2h^3,\,W+\delta W+2h^3) \end{aligned}
\end{equation}
\tag{2.4}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag \mathcal A[\Gamma_+]\ni A^+ &=(H_0^+,H_1^+,H_2^+,W^+) \\ &:=(2h^3,\,H_1+\delta H_1 -2h^3,\,H_2+\delta H_2+2h^3,\,W+\delta W). \end{aligned}
\end{equation}
\tag{2.5}
$$
To each of the points $A^\pm$ in coordinate spaces there corresponds a normalized branch divisor $\mathsf{E}\in\mathcal{H}_2^1$, containing two points $e$ in the upper half plane. Theorem. The displacement of a branch point $e$ caused by a tangential displacement $\delta A$ and a transversal displacement $h$ has the following asymptotic expansion: Remark 2. (i) We observe that the embedding of the coordinate space in the moduli space contains a root-like singularity near the appropriate part of the boundary of the space, $(H_0)^{2/3}$ for the space $\mathcal{A}[\Gamma_+]$ and $(W_2-W_1)^{2/3}$ for $\mathcal{A}[\Gamma_-]$; in particular, it is not continuously differentiable up to the boundary. (ii) The two integrals around the pole $z$ in (2.6) can be calculated explicitly using residues:
$$
\begin{equation}
\int_C d\eta^e=2\pi i\biggl(\frac{\Omega'(z)}{3\alpha^3}-\frac{4\beta^4\Omega(z)}{9\alpha^6}\biggr)\quad\text{and} \quad \int_C y(x)\,d\eta^e=2\pi i\frac{\Omega(z)}{3\alpha^2},
\end{equation}
\tag{2.7}
$$
where is the coefficient of the quadratic differential $d\eta_Md\eta^e$ and $\alpha>0$ and $\beta$ are coefficients of the expansion $\eta_M(x)=W+\alpha^3(x-z)^3+\beta^4(x-z)^4+\dotsb$. (iii) The machinery used to prove this theorem can be extended with minor changes to the description of the degeneration of a graph $\Gamma$ related to the merger of two zeros of the associated differential, as well as in the case of a higher genus $g$. Other types of degenerations like the merger of two branch points or of a zero of $d\eta$ and a branch point require a more significant modification of the method and will be addressed in our future publications. § 3. Proof of the main theoremThe plan of the proof is as follows: we consider the distinguished abelian integral on the unperturbed curve $M$ corresponding to a fixed point $A^0$ in the coordinate space of the unstable graph $\Gamma^0$. To make it single valued on the complex plane we have to introduce three cuts joining the branch points of the curve pairwise. Then, given sufficiently small tangential and transversal displacements $\delta A$ and $h$ we construct explicitly (a) a slight deformation $\eta^\pm(x)$ of the function $\eta_M$ and (b) a new global variable $\xi(x)$ in the complex plane, such that $\eta^\pm$ as a function of the new coordinate is the distinguished abelian integral on the modified curve $M^\pm$ corresponding to a point $A^\pm$ in the coordinate space of the stable graph $\Gamma_\pm$. Ahlfors’s formula (3.4) for an infinitesimal quasiconformal mapping (see, for instance, [22], Ch. V) will give us asymptotic formulae (2.6) for the displacements of the branch points $e\in\mathsf{E}$ of the curve. Now we proceed to the step-by-step realization of this plan.3.1. The abelian integralThe integral $\eta:=\eta_M$ of the distinguished differential on the curve $M$ corresponding to the point $A^0$ in the coordinate space $\mathcal{A}[\Gamma_0]$ of codimension $1$ is a locally single-valued function of $x$, except at infinity and on $\mathsf{E}$, where it has branch points. We introduce three disjoint cuts $B_j$, $j=0,1,2$, connecting the branch points pairwise, one of which passes through the point at infinity. We denote the two parts of it by $B_0^-$ and $B_0^+$, as in Figure 3, b. The abelian integral has a single-valued branch $\eta(x)$ in the remaining 3-connected domain (pants) in the complex plane of the variable $x$. Indeed, the homology basis of the pants (say the contours $C_1$ and $C_2$) can be lifted to even cycles on the curve $M$, which by definition survive the action of the reflection $\overline{J}$. The integral of the real differential $d\eta_M$ over an even cycle is real on the one hand (because of mirror symmetry) and purely imaginary on the other (because of normalization), hence it is zero. Note that the sum of boundary values of $\eta$ is locally constant on each cut since the distinguished differential $d\eta_M$ is odd with respect to the hyperelliptic involution $J$. These constants are purely imaginary because of the normalization of $d\eta_M$. They can easily be calculated since we can recover the values of the integral at all significant points of the curve, given the weights of the graph $\Gamma^0$ (this calculation requires a careful consideration of signs):
$$
\begin{equation}
\begin{gathered} \, \eta(e_1)=\eta(\overline{e_1})=iH_1, \qquad \eta(e_2)=\eta(\overline{e_2})=-iH_2, \\ \eta(1)=0, \qquad \eta(z)=W, \qquad \eta(-1)=i\pi. \end{gathered}
\end{equation}
\tag{3.1}
$$
3.2. A deformation of the abelian integralWe consider two smooth deformations of the abelian integral $\eta$ caused by the displacement $\delta A:= (\delta H_1,\delta H_2,\delta W)$ in the space $\mathcal{A}[\Gamma_0]$ and the transversal displacement $h>0$, which correspond to the choice of sign $\pm$ in the following formula:
$$
\begin{equation}
\eta^\pm(x):=\eta(x) -i\sum_{s=1,2}(-1)^s\delta H_s\rho_s(x)+(\delta W \pm 3h^2y(x))\rho(x).
\end{equation}
\tag{3.2}
$$
Here $0\leqslant\rho_s(x)\leqslant1$ is a smooth real-valued cut-off function equal to $1$ in a neighbourhood of the arc $B_s$ and vanishing identically outside some larger neighbourhood of the same arc; $\rho(x)$ is a similar function which is equal to $1$ in a neighbourhood of the double zero $z$ of $d\eta_M$. The supports of different cut-off functions are disjoint. The real coordinate $y(x)=\alpha(x-z)+\dotsb$ in a neighbourhood of $z$ is specified in the formulation of our theorem. 3.3. A new global coordinateLet $\xi(x)=\xi(x;\delta A, h, \pm)$ be the solution of the Beltrami equation $\xi_{\overline{x}}=\mu(x)\xi_x$ with coefficient $\mu(x):=\eta^\pm_{\overline{x}}/\eta^\pm_x$ which is homeomorphic in the whole plane and fixes the three points $x=\pm1,\infty$. The support of $\mu$ consists of three ring domains surrounding the cuts $B_1$ and $B_2$ and the double pole $z$ of $d\eta^e$. We assume without loss of generality that the values of each cut-off function $\rho_*(x)$ coincide at complex conjugate points. This implies that the Beltrami coefficient is mirror symmetric: $\mu(\overline{x})=\overline{\mu}(x)$. Since the normalizing set $x=\pm1,\infty$ of the map is real, the new variable $\xi(x)$ is mirror symmetric (real) too. 3.4. A key observationWe claim that the perturbed function $\eta^\pm(x(\xi))$ regarded as a function of the new global variable $\xi$ is the distinguished abelian integral for some disturbed curve $M^\pm$ with the branch divisor $\mathsf{E}^\pm:=\xi(\mathsf{E};\delta A,h,\pm)$ depending parametrically on the displacements. First we check that $\eta^\pm$ is a holomorphic function of $\xi$ outside the system of cuts $\xi(B_s)$, $s=0,1,2$. Indeed, the inverse mapping $x(\xi)$ is also quasiconformal with Beltrami coefficient $\nu(\xi)$ satisfying the relation (see [22]), which is obtained by differentiating the identity $\xi(x(\xi))=\xi$. Now
$$
\begin{equation*}
\eta^\pm_{\overline{\xi}}=\eta^\pm_x x_{\overline{\xi}}+\eta^\pm_{\overline{x}} \overline{x}_{\overline{\xi}}= \eta^\pm_x(x_{\overline{\xi}}+\mu(x)\overline{x}_{\overline{\xi}})=\eta^\pm_x(x_{\overline{\xi}}-\nu(\xi)x_\xi)=0.
\end{equation*}
\notag
$$
The next step is to check that the boundary values of $\eta^\pm$ on the sides of each cut add up to a purely imaginary constant, its own for every cut. This is clearly seen from (3.2): such a constant is equal to $-2i(H_s+\delta H_s)(-1)^s$ for the cut $B_s$, $s=1,2$, to $0$ for $B_0^+$ and to $i\pi$ for $B_0^-$. This observation means that $(d\eta^\pm)^2$ is a rational quadratic differential on the sphere of the variable $\xi$ with simple poles at the points in $\xi(\mathsf{E})$ and a double pole at infinity. The residue of $d\eta^\pm$ at infinity is the same as that of the differential $d\eta_M$ of the unperturbed curve. The fact that the constants found above are purely imaginary ensures the real normalization of the new differential. 3.5. Infinitesimal deformationWe use Ahlfors’s formula (see [22], Ch. V) for infinitesimal quasiconformal deformations to find the low-order terms in the deformation of the branch divisor $\mathsf{E}$. First we estimate the Beltrami coefficient by differentiating the explicit expression (3.2) for the deformation of the abelian integral and keeping only the leading terms of the deformation:
$$
\begin{equation}
\begin{gathered} \, \notag \eta^\pm_{\overline{x}}=-i\sum_{j=1,2}(-1)^j\delta H_j \rho_{j\overline{x}}(x) + (\delta W\pm 3h^2 y(x)) \rho_{\overline{x}}(x), \\ \notag \eta^\pm_x=\eta_x-i\sum_{j=1,2}(-1)^j\delta H_j \rho_{jx}(x) + (\delta W\pm 3h^2 y(x)) \rho_x(x)\pm3\rho h^2 y_x(x), \\ \mu(x)=-i\sum_{j=1,2}(-1)^j\frac{\rho_{j\overline{x}}(x)}{\eta_x}\delta H_j + \frac{\rho_{\overline{x}}(x)}{\eta_x}(\delta W\pm3h^2y(x)) + O\bigl(|\delta A|^2+ h^4\,{+}\,|\delta A|h^2\bigr), \end{gathered}
\end{equation}
\tag{3.3}
$$
where the residual term is uniformly bounded on the support of $\mu$. For each branch point $e\in\mathsf{E}$ we have an expression for its displacement induced by the change of moduli in the unstable coordinate space and the transversal displacement to the stable space:
$$
\begin{equation}
2\pi i\delta e= \int_{\operatorname{Supp}\mu} \frac{e^2-1}{x^2-1}\,\frac{\mu(x)}{x-e} \,dx\wedge\overline{dx} +O(\|\mu\|^2_\infty).
\end{equation}
\tag{3.4}
$$
Now we plug the approximate expression for $\mu(x)$ from (3.3) into this formula and use integration by parts, which reduces the formula to
$$
\begin{equation*}
2\pi i\delta e^\pm=\int_{\partial\operatorname{Supp}\mu} i\sum_{j=1,2}(-1)^j\delta H_j\rho_j(x)\,d\eta^e- (\delta W\pm3y(x)h^2)\rho(x)d\eta^e+O(\dots),
\end{equation*}
\notag
$$
where the meromorphic differential
$$
\begin{equation*}
d\eta^e:=(e^2-1)\frac{(x-\overline{e})(x-e')(x-\overline{e'})\,dx}{(x-z)^2w}, \qquad e'\in\mathsf E\setminus\{\pm1,e\},
\end{equation*}
\notag
$$
was defined in the statement of the theorem and the order of magnitude of the residual term is the same as in (3.3). The value of each mollifier $\rho_*$ is equal to $1$ on exactly one contour of the six that bound the support of the Beltrami coefficient and vanishes on the other five, and therefore the displacement of the branch point takes the form (2.6). 3.6. A new weighted graphIt is easy to draw the graph $\Gamma$ corresponding to the deformation (3.2) of the distinguished abelian integral in the unperturbed $x$-coordinate. The values of $W(x)=\operatorname{Re}\eta^\pm(x)$ have changed only in a neighbourhood of $z$, so the vertical part $\Gamma_ {\rule{.5mm}{2.mm}} $ of the graph remains intact. The horizontal part has been modified slightly since the deformation splits the double critical point $z$ into two simple ones. To find the zeros of the disturbed differential $d\eta^\pm$ we use the local coordinate $y$ in a neighbourhood of the double zero $z$ of the undisturbed differential $d\eta_M$. Setting $\rho=1$ in this neighbourhood we obtain the representation $\eta^\pm=W+\delta W+ y^3\pm 3h^2y$ for the integral. For sufficiently small $h>0$ the zeros $y=\pm h$ (for the deformation labelled ‘$-$’) or $y=\pm ih$ (for the deformation labelled ‘$+$’) of $d\eta^\pm$ lie in the same neighbourhood of the double zero with coordinate $y(z)=0$. Drawing the level lines of the function $\operatorname{Im}\eta^\pm$ that pass through the critical points found till their intersections with $\Gamma_ {\rule{.5mm}{2.mm}} $ we see that the graph associated with the new curve $M^\pm$ is (topologically) $\Gamma_\pm$. The critical values of the deformation of the abelian integral, together with its values at other significant points, allow us to recover the weights of the graph $\Gamma_\pm$. For example, consider the deformation ‘$+$’. For an arbitrary point $A^+=(H_0^+,H_1^+, H_2^+,W^+)\in\mathcal{A}[\Gamma_+]$ the critical values of the function $\eta_M^+$, which is defined outside a system of cuts homotopic to the $B_s$, $s=0,1,2$, are equal to $W^+\pm iH_0^+$. The values of $\eta_M^+$ at the branch points $e_1$ and $e_2$ are $i(H_0^++H_1^+)$ and $i(H_0^+-H_2^+)$, respectively; cf. (3.1). The calculations take into account the fact that branches of the distinguished differential outside the graph $\Gamma_+$ and outside the system of cuts $B_s$ differ at most in sign. The comparison with values taken explicitly from (3.2), for example, the critical values $\eta^+(\pm ih)=W+\delta W\pm 2ih^3$, shows that the new curve $M^+$ has the coordinates
$$
\begin{equation*}
A^+=(2h^3,\, H_1+\delta H_1 -2h^3, \,H_2+\delta H_2+2h^3,\, W+\delta W)
\end{equation*}
\notag
$$
in the space $\mathcal{A}[\Gamma_+]$. Similar calculations for the case of the deformation ‘$-$’ bring us to the point (2.4) in the coordinate space $\mathcal{A}[\Gamma_-]$. The theorem is proved. § 4. ConclusionWe have investigated the dependence of branch points of a curve on the length of the vanishing edge of a graph describing the conformal structure. To find the asymptotic behaviour of the embedding of the polyhedra $\mathcal{A}[\Gamma^\pm]$ in a neighbourhood of the face corresponding to the vanishing edge, we have adapted the technology of quasiconformal deformation of abelian integrals elaborated in [14], Ch. 5. It turns out that the dependence has a ‘cuspidal’ singularity in the vanishing transversal coordinate of the polyhedron and is smooth in all tangential coordinates. All constants in the asymptotic obtained are presented in an explicit controllable form as periods of a certain abelian integral. Higher-order terms of the expansion (2.6) can also be explicitly calculated; they will be presented elsewhere. Similar expansions can be produced in neighbourhoods of the ‘exterior’ faces of the coordinate space $\mathcal{A}[\Gamma_\pm]$ which correspond to the merger of two branch points.
Citation in format AMSBIB
\Bibitem{Bog23} |
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